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Atypical transistor-based chaotic oscillators: Design, realization, and diversityLudovico Minati, Mattia Frasca, Paweł Oświȩcimka, Luca Faes, and Stanisław Drożdż
Citation: Chaos 27, 073113 (2017); doi: 10.1063/1.4994815View online: http://dx.doi.org/10.1063/1.4994815View Table of Contents: http://aip.scitation.org/toc/cha/27/7Published by the American Institute of Physics
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Atypical transistor-based chaotic oscillators: Design, realization,and diversity
Ludovico Minati,1,2,a) Mattia Frasca,3 Paweł O�swiȩcimka,1 Luca Faes,4,5
and Stanisław Dro _zd _z1,61Complex Systems Theory Department, Institute of Nuclear Physics Polish Academy of Sciences (IFJ-PAN),Krak�ow, Poland2Centre for Mind/Brain Science (CIMeC), University of Trento, Trento, Italy3Department of Electrical Electronic and Computer Engineering (DIEEI), University of Catania, Catania,Italy4Healthcare Research and Innovation Program, Foundation Bruno Kessler (FBK), Trento, Italy5BIOTech, Department of Industrial Engineering, University of Trento, Trento, Italy6Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, Krak�ow, Poland
(Received 17 May 2017; accepted 7 July 2017; published online 21 July 2017)
In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar
junction transistors and a limited number of passive components can be obtained via random
search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after
manual adjustment of a variable resistor placed in series with the supply voltage source. Following
this approach, 49 unique circuits generating chaotic signals when physically realized were
designed, representing the largest collection of circuits of this kind to date. These circuits are
atypical as they do not trivially map onto known topologies or variations thereof. They feature
diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently
found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which gen-
erates a range of attractors depending on the parameter values. We also found a circuit yielding an
irregular quantized spike-train resembling some aspects of neural discharge and another one gener-
ating a double-scroll attractor, which represent the smallest known transistor-based embodiments
of these behaviors. Through three representative examples, we additionally show that diffusive
coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as
lag synchronization with directed information transfer and generalized synchronization. The repli-
cability and reproducibility of the experimental findings are good. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.4994815]
Transistor-based oscillators have been a ubiquitous sta-ple of electronics for decades, generating periodic signalsin disparate applications, e.g., communications, timing,and sound generation. It has been established that smallcircuits comprising at most few transistors can also gen-erate chaotic signals, which have complex features andare inherently unpredictable, though not random.Among other reasons, such chaotic oscillators haveattracted interest for their ability to replicate some phe-nomena occurring in biological systems when intercon-nected in networks. However, to date surprisingly little isknown about how to obtain them, even whether they rep-resent “unusual” or “special” situations. Here, a largenumber of transistor-based chaotic oscillators were auto-matically designed. These circuits do not trivially repre-sent known topologies, or variations thereof, and aretherefore “atypical.” They were physically built, thenstudied in terms of their overall features and certaincases of particular interest. Despite their simplicity, theygenerated a diverse range of signals and behaviors,including some typically associated with other systems.
The circuit diagrams and signals from all of them areprovided, considerably expanding the available reper-toire of oscillators of this kind.
I. INTRODUCTION
Countless low-order continuous-time systems exhibit
chaos for certain combinations of parameter values; new
examples are continuously identified,1 including recent
advances in systems with lines of equilibria,2,3 hidden attrac-
tors,4,5 and memristors.6,7 While the underlying nonlinear-
ities often involve polynomial terms or products of the
state variables, many functions are suitable for obtaining
chaos.8 Mathematical models can be transformed into analog
circuits following a consolidated approach9,10 wherein each
state variable is associated with a physical circuit quantity
(e.g., voltage across a capacitor or current through an induc-
tor), and the nonlinearity is realized exploiting the character-
istics of a semiconductor device or approximated by piece-
wise linear functions. Efforts towards discovering or design-
ing simple chaotic systems and implementing them physi-
cally are significant.11 To the circuits obtained following
this approach, one should add many others which generate
a)Author to whom correspondence should be addressed: [email protected],
[email protected], and [email protected]. Tel.: þ39 335486 670. URL: http://www.lminati.it.
1054-1500/2017/27(7)/073113/13/$30.00 Published by AIP Publishing.27, 073113-1
CHAOS 27, 073113 (2017)
http://dx.doi.org/10.1063/1.4994815http://dx.doi.org/10.1063/1.4994815http://dx.doi.org/10.1063/1.4994815mailto:[email protected]:[email protected]:[email protected]://www.lminati.ithttp://crossmark.crossref.org/dialog/?doi=10.1063/1.4994815&domain=pdf&date_stamp=2017-07-21
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chaotic dynamics by design such as the well-known Chua’s
circuit,12–17 and those which can exhibit chaos as an unde-
sired or unexpected feature (e.g., power converters18 and
oscillators19). Chaotic circuits feature nonlinearities originat-
ing from diverse devices including diodes, varactors,20 oper-
ational amplifiers,21 ferroelectric components,22 and
memristors.23
Compared to the general abundance of known chaotic
oscillators, relatively few circuit topologies based on a small
number of discrete bipolar junction-transistors (BJTs),
including non-autonomous24,25 and autonomous26–30 cir-
cuits, have been reported to date. Discrete BJTs have served
as the fundamental staple of electronics for decades, and an
extensive repertoire of circuits has been developed to serve
signal generation, amplification, and processing functions,
for example, in radio and audio applications. Hence, this
paucity is surprising. Indeed, although there exist methodolo-
gies or, at least, guidelines to design chaotic circuits based,
for instance, on the interaction of active networks and pas-
sive nonlinear devices,14 on time-delay systems,16 and on
operational amplifiers,31 we are not aware of general techni-
ques yielding circuits based on a limited number of discrete
BJTs as the only source of nonlinearity. The instances of
such circuits reported in the literature have been arrived at
either by serendipity or by following specific considerations:
many stem from modifications of existing periodic oscilla-
tors (e.g., the Colpitts oscillator,26 the Hartley oscillator,27
the blocking oscillator,28 and the inductor-resistance-diode
circuit24), or implement particular principles such as distur-
bance of integration, which underlies the non-autonomous
Lindberg–Murali–Tamasevicius (LMT) chaotic circuit.25
A preliminary study addressing this issue attempted to
use genetic algorithms, wherein evolution was driven
towards obtaining high-entropy signals in SPICE simula-
tions, and arbitrary circuit topologies were searched for by
representing connections and component values as a bit-
string.32 A number of autonomous chaotic oscillators were
successfully obtained, some of which were subsequently
characterized experimentally.30 One of them, featuring par-
ticularly small size (1 BJT, 2 inductors and 1 capacitor), was
later used as a building block to realize large networks,
which were able to replicate some emergent phenomena
originally observed in biological neural systems.33,34 That
study, however, had severe limitations. First, it did not
address to what extent the evolutionary aspect of the genetic
algorithm was significant, compared to the random search
component introduced by random cross-over and finite muta-
tion probability. When circuit topology and component val-
ues are conjointly represented, the majority of individuals
resulting from cross-over are expected to be structurally
invalid or inactive oscillators. Second, the individual fitness
was determined based on SPICE simulations, whose level of
agreement with experimental measurements had not been
evaluated; indeed, several studies have indicated that it can
be poor for chaotic oscillators of this kind.30,35,36 The accu-
racy of such simulations was certainly also constrained by
the fact that the circuits were physically realized by means
of low-end commercially-available components, associated
with significant parasitics and simplified models. The study
introduced, nevertheless, the useful notion of connecting a
variable resistor in series to the fixed DC supply voltage
powering the circuits, and using it as a control parameter to
manually search for chaotic ranges; owing to this, it was pos-
sible to obtain chaotic dynamics with diverse properties in
the physical realizations of the circuits.32
Here, we report on a multitude of chaotic oscillators
based on a limited number of passive components alongside
one or two BJTs as the only source of nonlinearity. These
circuits were obtained by means of a random search over the
space of 285 possibilities according to a bit-string represent-
ing discrete component values, in terms of a catalog of com-
mercially-available devices, and the connections between
them. Heuristic rules were applied to substantially reduce the
search space by excluding invalid individuals. One hundred
circuits were chosen based on SPICE simulations, physically
realized with state-of-the-art components, and experimen-
tally characterized. We illustrate their overall characteristics
with a focus on some cases of particular interest. In addition
to expanding the available repertoire of oscillator circuits of
this kind, this work posits that there is nothing “special” or
“unusual” about BJT-based chaotic oscillators; on the con-
trary, chaoticity is a common occurrence in valid BJT-based
oscillator circuits.
II. OSCILLATOR DESIGN AND REALIZATION
A. Search and simulation
Similar to Ref. 32, each oscillator was encoded as a ran-
dom string of 85 bits, allocated as shown in Fig. 1(a) to rep-
resent a circuit with up to 8 nodes (including 5 V DC supply
via resistor, and ground), 1 supply series resistor (value R, 16value steps in R¼ 464…2150 X), 6 inductors or capacitors(1 bit determining type, value L or C, 8 values of each inC¼ 150…1000 pF and L¼ 15…220 lH), and 2 BJTs offixed NPN type. The circuit was represented as a graph and
iteratively pruned until convergence, eliminating those ele-
ments that (i) had multiple terminals connected to the same
node (i.e., short-circuited components), or (ii) did not have
paths to both ground and supply nodes either directly or via
other elements, or (iii) had terminals not connected to at least
another component. Moreover, if (i) the circuit contained
fewer than two elements with connections to the ground and
supply nodes, or (ii) there was a path between the supply and
ground nodes comprising inductors only, or (iii) the supply
series resistor had been eliminated, the pruning process was
also canceled, and the individual discarded. If the process
completed and the circuit contained at least 1 inductor, 1
capacitor, and 1 BJT, the circuit was considered “valid” and
the corresponding SPICE netlist was written; otherwise it
was rejected as “invalid.” With the exception of not allowing
BJTs connected as diodes, effectively these rules did not
alter the outcome of the random search, only accelerating the
process by eliminating outright individuals that could not
oscillate in SPICE simulation. Similar search approaches
have been applied to other areas of electronic circuit design,
including layout optimization.37,38
An initial SPICE simulation was performed for each
valid circuit, running to 200 ls with a maximum step of
073113-2 Minati et al. Chaos 27, 073113 (2017)
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10 ns, ramping up the supply voltage during 100 ls, andmaintaining default trtol and reltol settings. The time-series segments for t> 175 ls were detrended with a 3rd-degree polynomial, and oscillations were deemed present if
resulting peak-to-peak voltage values ranged >100 mV.Circuits with at last one node meeting such a criterion were
deemed “active,” the others were rejected as “inactive.” A
second SPICE simulation was performed for each active cir-
cuit, running to 1 ms with a maximum step of 1 ns and reduc-
ing reltol to 0.0001.The time-series for t> 500 ls were linearly interpolated
to produce a fixed sampling time-series with a sampling
interval of 1 ns. The “dominant period” was found based on
the cross-correlation function, and the data were windowed
and resampled to yield 10 000 points within 50 times this
interval. The resulting time-series were cut into 4 segments
of 2500 points, and if the peak-to-peak voltage value was
>100 mV, the correlation dimension D2 was calculated foreach segment, using the Grassberger-Procaccia method
implemented as in Ref. 39, applying non-linear noise reduc-
tion with an embedding dimension s/2, and setting time-delay embedding s to the first minimum of the lag mutualinformation function, embedding dimension m to the mini-mum value yielding
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minpts ¼ 20 automatically identified two clusters, corre-sponding to periodic (n¼ 178, D2¼ 1.05 6 0.06) and chaotic(n¼ 117, D2¼ 2.10 6 0.21) signals, alongside a smallernumber of unclassified signals (n¼ 104, e.g., due to poorconvergence of the correlation dimension curves in the cho-
sen sampling window, or intermittency). As shown in Fig.
2(b), the chaotic signals had a lower spectral centroid44
(f¼ 5.10 6 4.68 vs. 9.05 6 4.16 MHz), reflecting greater gen-eration of slow fluctuations and, correspondingly, markedly
higher amplitude spectral flatness45 (n¼ 0.57 6 0.20 vs.0.08 6 0.03, considered range 5%–95% for f< 10 MHz).Assuming as boundary the maximum flatness observed in
periodic signals after rejection of one outlier, n< 0.27, 18signals misclassified by DBSCAN as chaotic were found to
be actually quasi-periodic, and accordingly had a comb-like
spectrum.46 The characteristics of all signals are detailed in
supplementary material Table II. Altogether, these experi-
mental results confirm that it is possible to obtain novel BJT-
based chaotic oscillators of diverse circuit topology and fea-
tures, based on a simple random search with suitable
heuristics.
Multifractal detrended fluctuation analysis (MFDFA)
over q¼ –4…4 with a detrending order m¼ 2 was subse-quently performed to determine the singularity spectrum f(a)on the chaotic time-series, extracting separately minima and
maxima to query the structure of amplitude fluctuations.47–49
There were 30 signals for which the width Da> 0.2 of f(a)suggests potential multifractality, but this was rejected in all
cases by consideration of the reshuffled and phase-
randomized time-series (also for larger range of q, data not
shown).50 This result indicates that even though chaoticity
was pervasive, the circuits did not produce multifractal
dynamics. Due to limited time-series length (median 1200
pts., analysis omitted if
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involves a complex relationship between circuit structure
and component values is reinforced.10 The circuits generat-
ing at least one chaotic signal according to DBSCAN (49 in
total) comprised 1 (1–4; median, range) capacitor, 3 (2–5)
inductors, 2 (1–2) BJTs, 6 (4–8) components excluding the
resistor, 5 (4–7) nodes, and 4 (1–8) LC combinations.
The full circuit diagrams of the 49 identified chaotic
oscillators, with associated waveforms, spectra, and attrac-
tors, are provided as supplementary material.
B. Simulation and measurement reliability
SPICE simulations had good accuracy predicting signal
amplitude (measured as voltage inter-quartile range, r¼ 0.75and r0 ¼ 0:63, where r and r0 denote, respectively, rank-order correlation between simulation and measurement
before and after manual resistor adjustment) and spectral
centroid f (r¼ 0.71 and r0 ¼ 0:69); however, they were poorat predicting chaoticity, as indicated by weak correlation
between simulations and experiment for both spectral flat-
ness n (r¼ 0.14 and r0 ¼ 0:17) and correlation dimension D2(r¼ 0.07 and r0 ¼ 0:17). This situation is in agreement withprevious observations for a chaotic Colpitts oscillator,35 the
LMT circuit,36 and other “atypical” circuits,30 but this study
is the first to systematically consider the issue of the accu-
racy of SPICE simulations in a large set of BJT-based cha-
otic circuits. While detailed investigation of this
disagreement is beyond the scope of this work, we performed
additional simulations with more stringent tolerance and step
settings, and rerun correlation dimension estimates for
SPICE waveforms attempting to closely replicate the experi-
mental settings (fixed sample rate, filtering 6 months: small errors for amplitude
(median absolute 6.3 mV, relative 1.7%), centroid frequency
f (43.1 kHz, 0.8%), spectral flatness n (0.012, 2.8%), and cor-relation dimension D2 (0.07, 3.1%) indicated that it was verygood. Second, measurement reproducibility was assessed by
building a second specimen of each oscillator from different
components: errors were larger but still small, for amplitude
(25 mV, relative 5.8%), centroid frequency f (244.3 kHz,3.1%), spectral flatness n (0.021, 5.3%), and correlationdimension D2 (0.12, 6.3%), indicating that it was also good.For 8 of these second specimens, as a consequence of com-
ponent tolerances, series resistor readjustment was necessary
to obtain chaoticity (jDR0j median absolute 142 X, relative14.0%). Third, the loading effect of attaching the oscillo-
scope probe (approximately C¼ 3.9 pF, Rp¼ 106 X,Rs¼ 150 X) was assessed by repeating all acquisitions multi-ple times, each time connecting a third probe to another
node, and considering the worst-case deviation. Errors were
considerably larger for amplitude (56 mV, relative 13.7%),
centroid frequency f (717.0 kHz, 13.5%), spectral flatness n(0.111, 39.8%). and correlation dimension D2 (0.22, 10.3%),indicating that, at least for some oscillators, the effect of
loading was more important than other error sources.
Altogether, these results (summarized in supplementary
material Fig. S1) reveal that despite the 610% tolerance inthe capacitor and inductor parameters and 630% tolerancein BJT parameters such as hFE, the reliability of the experi-mental findings was overall good, conferring practical value
to the experimental results on the present extended collection
of oscillators.
C. Smallest-size and other representative circuits
The smallest-size chaotic oscillators, which required 4
components in addition to the resistor, consistently featured
the topology shown in Fig. 3(a), which comprised 2 inductors
and 1 capacitor. The inductors connected the supply node to
the base and collector of the BJT, which rendered the currents
through them inter-dependent. Remarkably, in the random
search, this circuit topology recurred 7 times with only mini-
mal variations (e.g., BJT orientation) but different values of
the inductors and the capacitor (Table I). As a function of such
values, upon time-delay embedding of the experimentally mea-
sured signals, diverse dynamics were observed, including spi-
ral, phase-coherent attractors [Figs. 3(b), 3(d), and 3(e)],
attractors resembling the R€ossler funnel attractor [Figs. 3(c)and 3(g)], and attractors associated with a spiking behavior
[Figs. 3(f) and 3(h)].10,46 Both period-doubling and quasi-
periodicity (possibly favored by the presence of LC tanks with
mismatched frequencies) route-to-chaos were observed in
these circuits, a result that is relevant as quasi-periodicity has
been previously suggested as the prevalent route-to-chaos
mechanism in transistor-based chaotic circuits.30
The smallest-size BJT circuit previously studied in Refs.
30 and 32 was not identified, possibly due to the more
restricted component value ranges considered here, which
excluded its indicated inductor and capacitor values;
however, two variations of this circuit, including extra ele-
ments, were indeed found [circuits no. 21 and 23, see Figs.
4(b) and 4(c)].
073113-5 Minati et al. Chaos 27, 073113 (2017)
ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-27-012707ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-27-012707ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-27-012707
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Considering the circuits with 5 components in addition
to the resistor, 9 occurrences were found, in this case with
heterogeneous topology. Two of them [circuits no. 9 and 69;
Fig. 4(a) and supplementary material Figs. S2(a) and S2(f)]
represented a variation of the smallest-size topology
obtained by “degenerating” the supply or ground node by
means of an additional inductor. A further one [no. 34; sup-
plementary material Fig. S2(c)] was similar, with the addi-
tional inductor providing an extra tap to the supply node.
Two others [no. 42 and 90, supplementary Figs. S2(d) and
S2(i)] were structurally identical and contained two series
inductors; the remaining ones were all different. None repre-
sented a straightforward variation of a known circuit topol-
ogy, and considerable diversity of attractors was again
observed.
Further representative examples of the dynamics and
diversity observed in the larger circuits are shown in Fig. 4.
All spectra were characterized by a dominant component
surrounded by a multitude of narrow-band peaks [e.g., Figs.
4(b), 4(c), and 4(f)] and varying intensity of broad-band con-
tent [e.g., Figs. 4(a), 4(d), and 4(e)]. Instances of R€ossler-like funnel attractors [Figs. 4(a) and 4(f)], attractors similar
to the one of the Colpitts oscillator26 [Fig. 4(c)], reminiscent
of Shilnikov chaos53,54 [Fig. 4(d)], and an attractor with a
peculiar triple butterfly like shape [Fig. 4(e)] were observed;
instances of quasi-periodicity were also observed [Fig. 4(b)].
In all cases, close overlap in the spectra and attractors from
two different oscillator specimens was observed, confirming
good reliability of the experimental results.
D. Two notable circuits
Two further circuits demonstrated dynamics that were
particularly noteworthy. The first one (circuit no. 54), shown
in Fig. 5(a), consisted of two cascaded BTJs overlaid to a
network of 3 inductors in series and 1 capacitor, which pro-
vided multiple feedback paths. While at one node it gener-
ated a R€ossler-like funnel attractor [Fig. 4(f)], at anothernode this circuit generated activity resembling a bursting
spike-train, with spikes (i.e., impulses of approximately
quantized height) appearing as positive fluctuation followed
by smaller negative undershoot, having duration �0.25 ls[Fig. 5(b)]. All-or-nothing response was confirmed consider-
ing the distribution of local maxima amplitudes (
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network57 receiving two consecutive IEIs as input and hav-
ing 10 hidden neurons could predict the next IEI until �3 ls[rank-order r >0.85, Fig. 5(g), for representative example].Altogether, these findings indicate that the oscillator has all-
or-nothing dynamics which at the surface recall those of neu-
ral action potentials:56 however, even though the spike trains
qualitatively resembled bursts (avalanches), the dynamics
were not critical and the over-dispersion hallmarking “true”
burstiness was not present.58 Chaotic variants of BJT-based
blocking oscillators, which by their nature generate brief
pulses, have been proposed for broadband signal genera-
tion,28 and generation of bursts of pulses has been
observed;59 however, to the authors’ knowledge, the present
circuit does not represent a variation of a known topology. In
particular, oscillators generating a quantized response have
been previously described,61 but to the authors’ knowledge,
this is the first report of an autonomous BJT-based oscillator
with this behaviour; quantized spiking and bursting are more
often observed in complex circuits intentionally developed
as electronic models of neural dynamics.60 Future work
should explore the possibility of rendering this oscillator crit-
ical. Also in this case, there was a good overlap between two
specimens realized from different components.
The second circuit (no. 81), shown in Fig. 6(a), con-
sisted of two BJTs with a junction connected in anti-parallel,
2 inductors and 1 capacitor. Separate consideration of the
time-series at nodes 3 and 4 indicated that this oscillator
combined generation of continuous irregular activity with
switching between two unstable foci, resulting in a double-
scroll attractor [Figs. 6(b) and 6(c)]. Besides a complex spec-
trum featuring a large number of resonances overlapped to
broad activity [Fig. 6(d)], voltage at node 4 of this circuit
revealed a clear asymmetric double-scroll attractor [Fig.
6(e)] which was visible not only with time-delay embedding
but also when plotted with respect to voltage at node 3 [Fig.
6(f)]. Also for this oscillator, agreement between two
FIG. 4. Selection of oscillators having heterogeneous size, temporal, and spectral features: (a) circuit with 5 components, in addition to the resistor; (b), (c),
(d), and (f) circuits with 6 components; (e) circuit with 7 components. Replicability was confirmed by a close overlap between measurements from two physi-
cal specimens (blue, red; different series resistor values where indicated).
073113-7 Minati et al. Chaos 27, 073113 (2017)
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realized specimens was good. The double-scroll attractor is
characteristic of Chua’s circuit, a paradigmatic circuit that
has been realized in many distinct ways, including opera-
tional amplifier-based nonlinearities, inductor-less imple-
mentations, cellular neural network layouts, and monolithic
designs; these implementations start from the system equa-
tions and subsequently implement the characteristic function
of the Chua’s diode in several ways, or reinterpret the state
variables to obtain an equivalent circuit.9 Here, the circuit
was not been intentionally designed to produce a double-
scroll chaotic attractor, but rather was obtained by serendip-
ity. To the authors’ knowledge, it is the simplest known
BJT-based circuit producing a double-scroll attractor.
Another embodiment of Chua’s circuit only using 2 BJTs as
active elements has been previously described; however, it
also includes 7 resistors, 2 diodes, 2 capacitors and 1 induc-
tor (total 15 elements).62 More recently, an inductorless
double-scroll chaotic oscillator has been proposed; however,
since it is based on the RC phase shift, it also requires a large
number of components in addition to the 2 BJTs, namely, 7
resistors and 4 capacitors (total 13 elements).63 The present
circuit requires less than half the number of components
(total 6 elements), and as such is particularly important
towards proving the generative potential of small BJT-based
oscillators.
E. Synchronization
While the synchronization of structurally different cha-
otic systems has been thoroughly studied, experimental data
on heterogeneous BJT-based oscillators are limited.64 Here,
three paradigmatic cases are shown based on arbitrarily cho-
sen circuit pairs, to demonstrate the capability of these cir-
cuits to yield complex synchronization phenomena when
coupled into network configurations.
Three oscillator pairs were physically realized on a dedi-
cated substrate inside a ceramic dual-in-line package, to
simultaneously show the potential for integration into a
hybrid module usable for realizing large networks. These
modules included read-out amplifiers to minimize oscillator
loading (type MAX4200; Maxim Inc., San Jose, CA, USA).
A dedicated test board was developed, including additional
current adjustment (jointly for the two oscillators) and facili-
ties for hardware-based generation of bit-streams based on
maxima amplitudes for use as entropy sources; while this
feature was not utilized for the present study, its full design
FIG. 5. Spiking chaotic oscillator. (a) Circuit diagram. (b) Time-series recorded at node 3 in two physical specimens (blue, red; different series resistor values),
demonstrating replicable generation of spikes and bursts. (c) Distribution of maxima amplitudes vmax, confirming all-or-nothing behavior (dashed line: spikedetection threshold). (d) Distribution of inter-event intervals dt, revealing discrete steps. (e) Distribution of temporal Fano factor Ft for experimental andreshuffled series (dots, crosses), showing under-dispersion and absence of a power-law scaling region. (f) Corresponding distribution p(s) of avalanche size s,showing absence of heavy tail. (g) Scatter-plot between predicted d̂t and measured dt (nonlinear auto-regression) inter-event intervals for representative time-series, wherein strong correlation confirms deterministic dynamics.
073113-8 Minati et al. Chaos 27, 073113 (2017)
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is provided alongside that of the modules as supplementary
material. In all three cases considered the coupling was dif-
fusive, i.e., implemented by means of a resistor, with an
additional DC-blocking capacitor to avoid exchange of bias-
ing current. The coupling nodes, series supply, and coupling
resistor values were determined empirically. All raw time-
series are freely available.42
The first case, shown in Fig. 7(a), demonstrates the
emergence of lag synchronization, wherein oscillator X leadsY by d� 0.1 ls over a predominant period of �1.5 ls, asshown by time-lag normalized mutual information. To con-
firm asymmetric information flow, the transfer entropy was
estimated on 10 time-series pairs (25 000 pts.) according to
the method proposed in Ref. 65, which makes use of the
nearest-neighbor entropy estimator66 (implemented with
kneigh ¼ 10) and employs a non-uniform embedding tech-nique to limit the dimension of the variables involved in the
computation.67 The transfer entropy was considerably larger
in the X ! Y than in the Y ! X direction, with0.264 6 0.009 vs. 0.144 6 0.005, and the self-entropy wascomparable between X and Y, with 3.10 6 0.019 and3.27 6 0.02; the corresponding correlation dimension wasD2¼ 2.48 6 0.12 and 2.48 6 0.15, confirming chaoticity.While lag synchronization is a common observation in
weakly coupled heterogeneous oscillators,64 it does not
straightforwardly imply asymmetric information flow in the
direction of the lag. It has been shown that measures such as
delayed mutual information can be misleading in this
regard.68 Here, asymmetric information flow was confirmed
using transfer entropy, which overcomes the problems of
possible spurious detected coupling over uncoupled direc-
tions often encountered using time-delayed mutual
information.68,69
The second case, shown in Fig. 7(b), demonstrates the
achievement of near-complete synchronization between
structurally different oscillators. The phase-coherence value
between the time-series for X and Y was 0.95 6 0.001, andthe corresponding maximum cross-correlation coefficient of
amplitude fluctuations extracted via Hilbert’s transform was
0.979 6 0.0001. The corresponding correlation dimensionvalues were D2¼ 2.53 6 0.20 and 2.51 6 0.35, confirmingthat near-complete synchronization could be attained without
incurring oscillation death or destroying chaoticity. This
demonstrates the possibility of obtaining an almost-invariant
manifold between the two continuous chaotic systems.64,70
The third case, shown in Fig. 7(c), demonstrates the pos-
sibility of obtaining generalized synchronization, wherein a
complex functional relationship of the form y(t)¼w(x(t)) isestablished, instead of one between the scalar time-series.
Induction and detection of generalized synchronization are
non-trivial problems, particularly when dealing with experi-
mental systems whose dynamics are often influenced by
small parametric variations and parasitics, meaning that ana-
lytical approaches are difficult to apply.64,71,72 We resorted
to a metric based on agnostically determining whether close-
ness in response space implies closeness in driving space:
the L-index73 was calculated for 10 time-series pairs (12 500pts. each), time-delay embedded according to lag s set to the
FIG. 6. Double-scroll chaotic oscillator. (a) Circuit diagram. (b) and (c) Time-series recorded from two physical specimens (blue, red; different series resistor
values), showing irregular amplitude fluctuations at node 4 and switching behavior at node 3. (d) Amplitude spectra recorded at node 4, demonstrating replica-
ble generation of multi-component, broad spectrum. (e) Time-lag attractor reconstruction revealing asymmetric double-scroll geometry, closely overlapping
between the specimens (recorded with oscilloscope probe attached to node 4 only). (f) Corresponding physical-variable attractor reconstruction in voltages at
nodes 3 and 4.
073113-9 Minati et al. Chaos 27, 073113 (2017)
ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-27-012707ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-27-012707
-
first minimum of mutual information, dimension m¼ 1…4and kneigh ¼ 10. The maximum L-value between directionswas considered, and steadily increased with embedding
dimension m, reflecting representation of increasingly com-plex synchronization manifold geometry: synchronization
was weak for m¼ 1 (0.144 6 0.006), substantially increasedfor m¼ 2 (0.752 6 0.014), increased further for m¼ 3(0.905 6 0.002), and reached a level effectively indicatingcomplete synchronization for m¼ 4 (0.951 6 0.007). Thecorresponding correlation dimension values were D2¼ 2.796 0.16 and 2.79 6 0.28, confirming chaoticity. A similar sit-uation was also observed for the circuit in Fig. 7(a).
Generalized synchronization has been obtained numerically
and experimentally for diverse nonlinear electronic circuits,
mostly based on operational amplifiers, unidirectional cou-
pling, or other more complex arrangements.71,74–77 To the
authors’ knowledge, this is the first experimental observation
in which it spontaneously emerged between two diffusely
coupled BJT-based oscillators.
Multifractal detrended fluctuation analysis (MFDFA)
was also applied to the time-series of minima and maxima
generated by these coupled oscillators under a variety of set-
tings, but no evidence of multifractality was obtained (data
not shown).
IV. CONCLUSIONS
The results presented in this work highlight that chaos is
a common occurrence in small electronic circuits wherein
the only sources of non-linearity are the v–i characteristics in
the bipolar junction-transistor(s), and wherein self-sustained
oscillation is present. In the circuits considered here, the
probability of observing chaos was enhanced through instan-
tiating a variable resistor connected in series to the DC sup-
ply voltage, whose resistance represented the main control
parameter and was purposefully adjusted searching for cha-
otic ranges. The observation of chaos in approximately half
of the realized circuits (notably preselected among those
who featured self-sustained oscillation), despite the limited
accuracy of simulations in predicting chaoticity, indicates
that chaotic oscillators of this kind are definitely neither
“uncommon” nor “special”; considering them as such would
only be a consequence of the fact that a formal synthesis
method is still missing, and that the limited existing literature
focuses on chaotic adaptations of canonical topologies or cir-
cuits discovered by serendipity.24–30
By contrast, critical phenomena were not detected.
Further, the dynamics provided no convincing instances of
multifractality; however, anti-persistent mono-fractal
dynamics were prevalent. There are thus profound differ-
ences with respect to many self-organized biological and
physical systems, which often dwell close to the point of
criticality and exhibit signatures of multifractality.48,49,78,79
In particular, criticality in electronic circuits has received
very limited attention thus far, but a recent study on a lattice
of glow lamps has demonstrated the possibility of eliciting
critical phenomena by external tuning, even in the absence
of opportunity for self-organization to drive dynamics
towards criticality.55 It should be noted that the target of
these simulations and experiments was obtaining chaoticity,
FIG. 7. Examples of coupled heterogeneous oscillators showing, in order, circuit diagram, physical realization, time-lag normalized mutual information plot,
and Lissajous figure. Configurations yielding (a) lag synchronization, (b) near-complete synchronization, and (c) generalized synchronization. vs< 5 V due toshared current-adjustment circuitry on test board.
073113-10 Minati et al. Chaos 27, 073113 (2017)
-
not criticality; hence, the proximity of phase-transition
points, where critical phenomena preferentially occur, was
not explored systematically.
Owing to the fact that chaos is a pervasive occurrence in
these circuits, atypical topologies of chaotic oscillators, pre-
viously unknown and not intentionally representing varia-
tions of existing ones, could be identified by means of a
random search process. The search process was made com-
putationally tractable by applying suitable heuristics, elimi-
nating a-priori invalid circuits without attempting tosimulate them, and then using a two-step approach to only
run time-consuming simulations for circuits already ascer-
tained to be oscillating; this “funnel” approach meant that
time-consuming simulations were run only a small fraction
of the initial candidates (�0.1%). While the usage of geneticalgorithms to design these circuits was previously advo-
cated,32 the present results suggest that maintaining a similar
bit-stream representation of circuit topology and parameters,
the search process can effectively be approximated by a ran-
dom one. Such assertion is supported by the fact that, when
crossing-over the genetic code of two different oscillators,
the majority of resulting circuits are not functioning oscilla-
tors, as observed for other situations where structures and
parameters are conjointly represented. Future work may re-
consider the use of genetic algorithms applied in a narrower
scope for the optimization of parameter values in these
circuits.37,38
Compared to preexisting work,30,32 significant instru-
mental improvements were introduced, attempting to
enhance the accuracy of SPICE simulations; in particular,
the physical inductors were modeled by means of realistic
RLC networks, high-grade capacitors and transistors with
low parasitics were chosen, and interconnections were real-
ized with optimized printed circuit boards. Nevertheless,
SPICE simulations could only predict with good accuracy
the amplitude and frequency centroid of the generated sig-
nals, but they were unsuccessful at predicting onset of chaos,
as indexed by uncoupled spectral flatness and correlation
dimension in comparison to the experimental data. Because
the reproducibility of the experimental results across circuit
specimens realized with different components was good,
such disagreement could not only be due to parametric mis-
matches between physical components and canonical values.
This result thus highlights limitations inherent in the compo-
nent models and numerical solver. A substantial number of
chaotic oscillators were nevertheless identified, owing to the
fact that (i) as stated above, chaoticity is a common occur-
rence in these circuits, (ii) these circuits included a supply
series resistor, which was intentionally adjusted searching
for chaotic ranges, (iii) even though the simulations failed to
predict chaoticity, they successfully delivered a set of atypi-
cal circuits all of which (except one) actually oscillated
when physically realized: this restricted the search to a very
small fraction of the entire set of hypothetical oscillators
described by the generated random bit-strings.
A recurrent topology was identified in circuits compris-
ing 4 elements in addition to the series resistor. To the
authors’ knowledge, it did not represent a previously known
oscillator, and it demonstrated a remarkable universality in
that the 7 instances found generated a range of diverse attrac-
tors as a function of the component values. The size of this
circuit is comparable to the smallest known autonomous
oscillators of this kind.26–28,30 Even though a formal route-
to-chaos analysis was not conducted, both period-doubling
cascade and quasi-periodicity were commonly observed and
identified as primary mechanisms to the onset of chaos in
these circuits.
Furthermore, two novel oscillators noteworthy for their
dynamics were obtained. The first one generated spikes
approximating an all-or-nothing (quantized) response.
Analyses of temporal scaling and predictability of inter-event
intervals indicated that the underlying dynamics were strongly
deterministic, and signatures of criticality such as power-law
scaling of avalanche size were missing. Nevertheless, this
oscillator is of particular interest, as it demonstrates the possi-
bility of observing chaoticity in transistor-based circuits in
the form of irregular inter-event times between spikes, rather
than cycle amplitude fluctuations. While a range of quantized
oscillators capable of generating spikes and bursts have
been described previously, these often implement integrate-
and-fire dynamics based either on considerably more complex
circuits aiming to mimic neurons60 or on highly non-linear
components such as glow lamps (gas discharge tubes).55 Even
though critical signatures were missing, there were elements
of qualitative similarity to neural discharge time-series; hence
future work should explore the possibility of modifying or
externally tuning this circuit to yield critical behavior.
The second oscillator generated an asymmetric double-
scroll attractor via combining irregular cycle amplitude fluc-
tuations and alternation between two unstable foci. There are
a variety of numerical and experimental systems, which can
give rise to this attractor, of which to the authors’ knowledge
only two implementations based on bipolar-junction transis-
tors are known.62,63 Compared to them, the circuit consid-
ered in this study is considerably smaller as it involves less
than half the total number of elements, namely, 2 inductors,
2 transistors, 1 capacitor, and 1 resistor, and as such, it rein-
forces the universality of these oscillators.
By means of three arbitrary but representative examples,
it was also demonstrated that structurally heterogeneous cir-
cuits can be diffusively coupled, giving rise to non-trivial
synchronization scenarios. While diffusive (resistive) cou-
pling is inherently symmetrical, it was shown that, depend-
ing on the oscillator dynamics, asymmetric inter-dependency
can emerge, hallmarked by lag synchronization and directed
information transfer between the two oscillators. It was also
demonstrated that in some cases an invariant synchronization
manifold may exist enabling near-complete synchronization
between different oscillators, without incurring in oscillation
death or loss of chaos. Furthermore, the possibility of sponta-
neous emergence of generalized synchronization was shown,
by means of applying a rank-based affinity metric, which
consistently increased up to four-dimensional embedding.
Taken together, these results demonstrate the potential of
these circuits to spontaneously generate non-trivial synchro-
nization phenomena, leading to network complexity when
coupled in heterogeneous ensembles, as is often the case in
natural systems; this is of broad interest since, to date,
073113-11 Minati et al. Chaos 27, 073113 (2017)
-
limited experimental research has been done on emergence
in networks of electronic oscillators mismatched structurally
rather than just parametrically.64,80,81
More generally, this study provides a large collection of
transistor-based chaotic circuits, with substantial diversity of
topological and dynamical features. All circuit diagrams and
experimental time-series are freely available, supporting
future research and applications in this area, particularly as
the findings were largely reproducible across realizations of
each circuit.
SUPPLEMENTARY MATERIAL
See supplementary material for additional tables, fig-
ures, circuit diagrams, signals, circuit board fabrication
materials, and illustrations.
ACKNOWLEDGMENTS
The authors are grateful to Gianluca Giustolisi for
insightful discussions into the operation principles of these
circuits, to TDK Corporation (Tokyo, Japan) for providing
realistic RLC network models of the inductors, and to
Tecno77 S.r.l. (Brendola VI, Italy) for assistance during
board layout design. All experimental activities were self-
funded by L.M. personally and conducted on own premises.
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